Discrete solitons in media with saturable nonlinearity
Doctoral thesis (Published version)
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Solitons represent localized structures which exist due to the exact balance between nonlinear self-interaction and dispersive and/or diffractive effects. They can be found in quite different nonlinear systems in nature such as fluids, optical fibers, Bose-Einstein condensates, etc. Mathematically, solitons are the exact particular solutions of various nonlinear partial differential equations, such as Korteweg-de Vries, Klein-Gordon, and Kadomtsev-Petviashvili equations. They have an infinite number of conserved quantities and demand an infinite phase space to exist. However, most nonlinear physical systems of importance are described with non-integrable evolution equations that have so-called solitary wave solutions. These localized structures, although with only few conserved quantities like power and Hamiltonian and in spite of radiation losses, exhibit a surprising robustness and vitality during their propagation and interactions. Especially in the optics community it is ...quite usual to neglect these differences between solitons and solitary waves and to use only the term soliton, which shall be performed through this thesis. The first scientifically documented report about the beautiful phenomena of soliton formation originates from the Scottish naval engineer John Scott Russell, who observed “a wave of translation” while riding a few miles after it on horse back along a narrow barge channel near Edinburgh in Scotland in August 1834. Years later after this first observation he built a tank in his own garden and started to experiment with shallow water waves. Russell discovered that their shape can be described by a sech2 function and determined that their peak amplitude is proportional to the velocity of the wave. However, at that time there was no equation describing such water waves and possessing solitary solutions. It took more than sixty years until Korteweg and de Vries in 1895 derived a nonlinear wave equation which describes the evolution of waves in a shallow one-dimensional (1D) water channel. As a confirmation of Russel’s experimental investigations they have shown theoretically that such a system admits solitary wave solutions. From that time, being regarded unstable for all possible initial conditions solitons stayed dormant for decades. 1960 Gardner and Morikawa have proved that these solutions are spatio-temporal stable structures for a wide set of initial conditions. Five years later Fermi, Pasta and Ulam have explored the mechanisms which lead to thermal equilibrium. They numerically integrated the ordinary differential equations which describe a set of coupled anharmonic oscillators. Because of the equipartition of energy they thought that the nonlinearity would quickly cause energy redistribution among all the modes, but they found that only a very small number of modes were actually participating in the system dynamics. In the same year Zabusky and Kruskal, while studying the Fermi-Pasta-Ulam problem, rederived the Korteweg-de Vries equation as a continuum approximation. They have numerically solved this equation for periodic boundary conditions and revealed that the solitary solutions of this equation interact elastically with each other. Exactly because of this particle-like property they named these solutions “solitons”. In 1967 Gardner, Greene, Kruskal and Miura, while exploring the initial value problem for the Korteweg-de Vries equation, discovered a new method of mathematical physics based on the ideas of direct and inverse scattering. One year later Lax generalized these ideas, and in 1971 Zakharov and Shabat proved that this method also can be applied to another physically significant nonlinear evolution equation, namely the nonlinear Schrödinger (NLS) equation. Three years later Ablowitz, Kaup, Newell and Segur showed that this method is analog to the Fourier transform for nonlinear problems. They called this procedure the inverse scattering transform. Although solitons are observed in various nonlinear systems such as DNA 1 molecules and Scheibe aggregates, in this thesis the accent is put on solitons in nonlinear optics. The realm of nonlinear optics consists of those phenomena where the optical properties of a material depend on the strength of the applied field. As only the laser light usually is intense enough to modify the optical properties of a dielectric material it is not surprising that the beginning of nonlinear optics is usually taken to be 1961, when Franken, Hill, Peters and Weinreich observed the generation of a second harmonic frequency by focusing a laser beam through crystalline quartz. The idea that an optical beam may induce a waveguide and guide itself in it was suggested by Askary’an one year later. Spatial self-focusing of optical beams due to third-order nonlinearities was for the first time experimentally analyzed in 1964, while the first experiment on spatial solitons was reported one year later by Ashkin and Bjorkholm. The existence of temporal optical solitons in lossless fibers was theoretically proposed by Hasegawa and Tappert in 1973, and the first experimental realization in a long-distance all-optical transmission line was carried out by Mollenauer and Smith in 1988. Discrete solitons, which are the main objects of investigation in this thesis, were theoretically suggested by Christodoulides and Joseph 16 years ago in 1988. The first experimental observation of these localized structures, which possess a great potential in optical communications, was published only recently in 1998. In this thesis the dynamics of discrete solitons in media with a saturable type of the nonlinearity is investigated analytically, numerically, and experimentally. A more detailed classification of solitons with respect to their dimensionality, colour, coherence, and nonlinear mechanism is performed in the second chapter. Chapter 3 describes the connection between discrete solitons in periodic waveguides and band gap structures in solid state physics, including the phenomena of discrete diffraction and Bloch oscillations. Furthermore, a usual approach for describing discrete solitons in the first band-gap, the so-called tight-binding approximation, is tackled. In this approximation the system dynamics is represented by the discrete NLS (DNLS) equation. It is shown that a plane wave solution of this equation is modulationaly unstable and it might evolve to a discrete soliton. Subsequently, a rough classification of different discrete solitons is presented. Finally, at the end of this chapter, the stability of solitons is investigated. In the next chapter the model equation, which describes the propagation of optical spatial pulses in discrete media with saturable screening photorefractive nonlinearity, is exposed. A few stationary unstaggered solutions are obtained and their stability with respect to small perturbations is investigated. The numerical examination of the stability of stationary soliton solutions reveals that such systems exhibit a cascade mechanism of saturation. A similar model equation which describes the propagation of optical pulses in discrete media with saturable photovoltaic, photorefractive nonlinearity is presented in Chapter 5. Also, the stability of different analytically discovered stationary staggered solutions is examined here. Chapter 6 is devoted to the intrinsically localized modes (i.e., discrete screening and photovoltaic solitons) and the Peierls-Nabarro (PN) potential barrier, which can be roughly described as the energy difference between a localized mode centred on the site (mode A) and a localized mode centred between sites (mode B). It is discovered that this potential, opposite to discrete media with cubic nonlinearity, may change its sign, which has a strong influence on the stability of the modes A and B. The corresponding PN potential has multiple zeroes, which enables increased mobility of discrete solitons across the system. New phenomena, such as stable propagation of mode B across the array and elastic interactions of discrete screening photorefractive solitons are presented. Chapter 7 is reserved for experimental results. The procedure of channel waveguide formation in lithium niobate (LN) and strontium barium niobate (SBN) is explained at the beginning of this chapter. Experimental observations of discrete diffraction, diffraction-less propagation in LN nonlinear waveguide arrays, formation of stable discrete photovoltaic solitons as well as their steering across the array are demonstrated. Conclusions and possible directions for future investigations are given in Chapter 8.
Keywords:solitons / discrete solitons / Peierls-Nabarro potential
- Technical University of Clausthal, Faculty of Mathematics and Natural Sciences